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On the summation of Taylor’s series on the contour of the domain of summability

  • Nikola Obrechkoff
Published/Copyright: November 11, 2016
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 19 Issue 5

Abstract

Editors’ notes [*]: In this paper N. Obrechkoff defined a general method of summation or, more precisely, a method of Borel (B) type for analytical continuation of complex function f defined by convergent series of the kindc0+c1z+c2z2+…. The main goal is the summation of such series at boundary points of the corresponding region of summability under conditions of the kind B for the singular points of the function f, as they have been introduced in his earlier paper [8], 1928.

Eαp(x)=∑n=0∞xnΓ(αn+p+1),0<α≤2,integerp≥0,asx→∞,(*)

which for p = 0 reduces to the well-known Mittag-Leffler functionEα (x). He obtained thatEαp(x)=βx−pβexβ+η(x),β=1α,for|argx|≤απ2

where η (x) → 0 as x → ∞. Note that in contemporary denotations, (*) is the 2-parameter Mittag-Leffler functionEα, μ (x) with μ = p + 1, known now as the Queen Function of Fractional Calculus (Gorenflo and Mainardi, 1997).

Further, the already proved theorems are specified for the Borel as well as for the Mittag-Leffler type summations. To this end, Obrechkoff studied the asymptotics of the function (see § 11)

MSC 2010: 33E12; 40G05; 40G10
Key Words and Phrases: Mittag-Leffler star; Mittag-Leffler function; summability of divergent series; special methods of summability – Abel; Borel; Cesàro; Euler-Knopp; Hausdorff methods

References

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[7] N. Obrechkoff, On the summation of divergent series. Annuaire Univ. Sofia, Phys.-Math. Fac. 24, No 1 (1928), 93–160 (see p. 147).Search in Google Scholar

[8] [added by Eds, now] N. Obrechkoff, Sur la sommation de la serie de Taylor sur le contour du polygone de sommabilite par la methode de M. Borel. C. R. Acad. Paris186 (1928), 1813–1815.Search in Google Scholar

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Published Online: 2016-11-11
Published in Print: 2016-10-01

© 2016 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 19–5–2016)
  4. Round Table Discussion
  5. Fractional Calculus: D’où venons-nous? Que sommes-nous? Où allons-nous?
  6. Survey Paper
  7. Models of dielectric relaxation based on completely monotone functions
  8. Survey Paper
  9. Space-time fractional stochastic equations on regular bounded open domains
  10. Discussion Paper
  11. Geometric interpretation of fractional-order derivative
  12. Survey Paper
  13. Fractional calculus in image processing: a review
  14. Research Paper
  15. Structural derivative based on inverse Mittag-Leffler function for modeling ultraslow diffusion
  16. Research Paper
  17. On the regional controllability of the sub-diffusion process with Caputo fractional derivative
  18. Discussion Survey
  19. There’s plenty of fractional at the bottom, I: Brownian motors and swimming microrobots
  20. Research Paper
  21. On a fractional differential inclusion with “maxima”
  22. Research Paper
  23. On time-fractional representation of an open system response
  24. Archive Paper
  25. On the summation of Taylor’s series on the contour of the domain of summability
https://doi.org/10.1515/fca-2016-0069
Keywords for this article
Mittag-Leffler star; Mittag-Leffler function; summability of divergent series; special methods of summability – Abel; Borel; Cesàro; Euler-Knopp; Hausdorff methods

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 19–5–2016)
  4. Round Table Discussion
  5. Fractional Calculus: D’où venons-nous? Que sommes-nous? Où allons-nous?
  6. Survey Paper
  7. Models of dielectric relaxation based on completely monotone functions
  8. Survey Paper
  9. Space-time fractional stochastic equations on regular bounded open domains
  10. Discussion Paper
  11. Geometric interpretation of fractional-order derivative
  12. Survey Paper
  13. Fractional calculus in image processing: a review
  14. Research Paper
  15. Structural derivative based on inverse Mittag-Leffler function for modeling ultraslow diffusion
  16. Research Paper
  17. On the regional controllability of the sub-diffusion process with Caputo fractional derivative
  18. Discussion Survey
  19. There’s plenty of fractional at the bottom, I: Brownian motors and swimming microrobots
  20. Research Paper
  21. On a fractional differential inclusion with “maxima”
  22. Research Paper
  23. On time-fractional representation of an open system response
  24. Archive Paper
  25. On the summation of Taylor’s series on the contour of the domain of summability
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